42. Whatever the value of p > 0 in the equation y = x²/(4p), the y-coordinate of the centroid of the parabolic segment shown here is ỹ = (3/5)a. a y х ||

Q: 3. Evaluate the given line integral. Follow the direction of C as given in the problem statement. 2.…

A: Line integral is integral of some function along a curve or path. Any field for which the line…

Q: 13 - x² – y? at (3, 2, 0), then If the point (1, 1, k) lies on the tangent plane to the paraboloid z…

A: z=13-x2-y2 Partially differentiating with respect to x, we get zx=0-2x-0=-2x Partially…

Q: 1 The domain of the function f(x,y)= + In(4- 4x2 - y?) is the region inside the x²+4y² – 4 ellipse…

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Q: 8.Evaluate the triple Integral. Te by The paraboloid X=2yt + 2z? and the plane X=2. 7x dV, where E…

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Q: B. Evaluate the given line integral. Follow the direction of C as given in the problem statement. 3.…

A: Given ∫Cz3-4x+2y dswhere C is the line segment from 2,4,-1 to 1,-1,0

Q: Evaluate the line integral y = x² from (-1, 1) to (1, 1) and the line segment from (1, 1) to (−1,…

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Q: 18. Show that the centroid of the sector in Figure 13 has y-coordinate 2R sin o (0, y) R FIGURE 13

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Q: 6. Evaluate the line integral g. x²y² dx + xy dy where C consists of the arc of the parabola y = x²…

A: (a) Direct method The given line integral is ∮C x2 y2 dx + xy dy. The curve C consists of    C1 :…

Q: 10. Evaluate the integral cz(z-1) where C is a simple closed curve in the positive direction that…

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Q: (b) 6xy dV, where E lies under the plane z=1+x+y and above the region in the xy-plane bounded by the…

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Fined Fhe boundled by Ahe Cure area 2. x+Y= 3 %3D -33-j` and ど= %3D Ans wer

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Q: Calculate the volue of the de integral 23 - 1 2 F soro fallowing c a) Cil2+1|= 1 Curves. 6)…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: The region in the first quadrant is bounded by the curve y = 2 - x^2 and y = x^2. How far is the…

A: Given- The region in the first quadrant is bounded by the curve y = 2 - x2 and y = x2.  To find- How…

Q: (2) Use the transformaiton u = xy, v = x² – y? to find JJ (æ* – y*)e*³ dA where R is the region in…

A: Given: The region bounded by the hyperbola xy=1 &3, x2-y2=3 &4 Where u=xy and v=x2-y2

Q: 1. Find the region in the xy-plane in which the following equation is hyperbolic: [(x-y) -1]u +2uy…

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Q: 17. Evaluate the line integral xe" dx +y sin yªdy along the triangular path consisting of the line…

A: Consider the given integral. I=∫Cxex2dx+ysiny2dy The given points are 0,0,2,3,1,0,0,0. Now, draw all…

Q: 10.18. Use an appropriate change of variables to evaluate the integral ff,(1 + x? + y²)²dA, where T…

A:

Q: 1. Compute SF · dr where F(x, y) = (y²,–x²) and C is the part of th parabola u = 12 from (-1 1) to…

A: First take dot product of f and dr Then put value of y in terms of x Integrate and raise limit of…

Q: 2 2 JEind. thePaints.an.th. sur face. Xy Z=1 that.are .... closest...t. the origin

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Q: 1. Set up the formulation to find the surface generated when the curve y = x² is revolved about the…

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Q: 1.13 Evaluate the line integral $ x*dx + xydx where C is the triangle with vertices at (0, 0), (0,…

A: The given function is x4dx+xydy By green's theorem, the integral becomes, ∫06∫06∂M∂y-∂N∂xdxdy where…

Q: 5) Evaluate the line integral S YX²dX + XY²dY on curve C, where C is the upper portion of the www m…

A:

Q: B. Evaluate the given line integral. Follow the direction of C as given in the problem statement. 4.…

A: Given ∫Cy2-3z+2 dsC is the line segment from 1,0,4 to 2,-1,1

Q: (b) || 6xy dV, where E lies under the plane z 1+x + y and above the region in the xy-plane bounded…

A: A triple integral involves the integral with respect to x,y, and z. In this problem, we have to…

Q: 8. Evaluate the line integral x dx + xedy where C is the closed path along the triangle in the…

A:

Q: 15. Evaluate the line integral J. (3xy* - 6xy ]dx + (4x'y - 3x) dy, where C is any path from (1, 2)…

A:

Q: 1. Evaluate the line integral of E = xax - y ay along the segment P₁ to P2 of the circular path as…

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Q: integrate ƒ over the given region. ƒ(s, t) = e^s ln t over the region in the first quad-rant of the…

A:

Q: 22. Evaluate xyz dV, where E lies between the spheres p - 2 and p- 4 and above the cone - 7/3.

A: Solution:-

Q: Evaluate the triple integral.  (x2 + y2) dx dy dz; W is the pyramid with top vertex at (0, 0, 2)…

A: The given triple integral is,

Q: 1. Use a change of variables to evaluate || ソーx dA where R is the trapezoid with vertices (1,0,…

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Q: у 3 1 + sin х, 0 <x< т/2 y

A: The given curve is y=1+sinx and the limit is 0≤x≤π2 The Surface area is given by S=∫x2πy1+y'2dx or…

Q: 1 Find the centroid of the area bounded by x2- 4y + 8 = 0, the lines x = 0 and x = 4, and the…

A: Formulas are written within the solution.

Q: 11 Calculate the line integral f Gy)ds for the given junc rion and path xiyl=xy KHt)=<3+,2t=-37 for

A: r(t)=&lt;3t,2t-3&gt;=&lt;0,-3&gt;+t&lt;3,2&gt; x(t)=3t  and y(t)=2t-3 ds=32+22dt=13dt 0≤t≤1

Q: Compute algebraically the resultant of the following coplanar displacements: 20.0 m at 30.0°, 40.0 m…

A: For the given data, the horizontal and vertical component is  So, the horizontal component is…

Q: (1) Find the area of the curved surface of the upper half of the sphere x2 + y² + z² = a²(a > 0) on…

A:

Q: 1 (#3) Find the surface area for y = 3 on [0, 3] rotating on the x- axis.

A: Given an equation of a curve, The solid will form by rotating the curve about the x-axis.

Q: 7. Evaluate the integral I | sin(9.r² + 4y²)dA by hand, using an appropriate change of variables,…

A: The given question can be solved as shown in step2

Q: 3 2 Use a line integral to calculate the length of the helix ř(t) = (a cost, a sint, bt) 0<t < 2n…

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Q: a Z2=2+4i - [ (3 Im (2) + 2 Re (2)] dz Let e = 21=0 where C the parabola y = x² defined from z1 to…

A: Solution: (a) Let us consider C be a parabola y=x2 defined from z1=0 to z2=2+4i. Therefore, C can be…

Q: 2. Suppose that the edge lengths r, y, and z of a closed rectangular box are changing at the…

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Q: Find the area of the surface generated when the curve y=1/4(e2x + e−2x)is rotated about the x-axis…

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Q: 1. Use a change of variables to evaluate [[ y- dA where R is the trapezoid with vertices (1,0,…

A:

Q: Obtain the Solution x, y > 0, xy > 1 of the PDE 2y on the x+y 1 Uxy such that u = x+y 0, ux…

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Q: 1 9. Evaluate dV;where E is the solid between the spheres of radius 1 (x² + y² + z²)³/2 E and 2…

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Q: Find a cubic approximation of r(θ) = (θ 2 , eθ , θ) at θ = 0.

A: given rθ=θ2,eθ,θ to find - cubic approximation at θ=0

Q: 1. Find parametrizations of the curve of intersection of x² + 4x + z² = 0 and x- – 2y + z = 3.

A: Given curve is x2 + 4x + z2 = 0 .  This can be written as (x+2)2 + z2  = 4  So the curve is a circle…

Q: 8. Evaluate the line integral (x'dx+ xe"dy where C is the closed path along the triangle in the…

A: Here we have to evaluate the integral ∫cx2dx+xe-ydy where C is the closed path along the triangle in…